Department of Mathematics,
Arizona State University
Tuesday,
April 14, 1997, 3:05 p.m. in PSA Room 306
Leigh J. Little
Department of Mathematics
A Finite Element Solver for the Navier-Stokes Equations using a
Preconditioned Adaptive BICGSTAB(L) Method
Abstract
The physical behavior of many fluids is
governed by the incompressible Navier-Stokes
equations. The equations are far too complex to
be computed analytically and hence numerical
solutions are necessary. To study the numerical
solution of these equations, a large computer
program, BCSLNSS, was written in Fortran-90.
This program solves the 2-dimensional, incompressible
Navier-Stokes equations in a primitive variable
formulation on polygonal Dirichlet domains.
Discretizations in space and time are performed
using the P1 Bubble-P1 finite element method and
a one-step Theta method respectively.
Krylov subspace methods are used to solve the
resulting linear systems. The conjugate gradient
method will usually not perform well because the
system matrices are indefinite. This leads
to the class of BICG methods. A recent member
in this class is the BICGSTAB(L) algorithm.
In this paper, a method called ASTAB(L)
is proposed. This algorithm takes advantage of the
fact that the value of L can be adaptively selected.
Experiments were performed with BCSLNSS to
assess the validity of the code. In addition, the ASTAB(L)
algorithm is compared to the BICGSTAB(L) algorithm
to see if any advantage is gained in the adaptive
strategy. The results indicate that the adaptive method
is generally not more efficient, but is safer to use.
Finally, experiments were performed to determine a
suitable preconditioners in solving the velocity and
pressure equations. It was found that the ILUT(m) preconditioner
is outstanding in comparison to simple diagonal preconditioners
for convection-diffusion solves. Selecting a suitable pressure
preconditioner is a difficult task because the coefficient
matrix is never available. In spite of this the application
of the ILUT(m) preconditioner to simple approximations of
the pressure matrix seem promising.
For further information please contact:
mittelmann@asu.edu